Functional forms in joint models for longitudinal and time-to-event data: A practical guide with application and interpretation

Abstract

Background: Joint models for longitudinal and time-to-event data are widely used in clinical research. However, the choice of functional form linking the biomarker trajectory to event risk is often treated as a technical detail, despite its importance for model assumptions and interpretation. Default specifications may fail to capture clinically relevant features of biomarker trajectories. Methods: We provide a structured overview of functional forms linking longitudinal and survival processes in joint models. We compare association structures including instantaneous effects (current value, slope, and acceleration), cumulative and change-based formulations, shared random effects, and variability-based associations. Using longitudinal white blood cell measurements and overall survival data from the MIRAGE glioblastoma trial, we illustrate how different functional forms capture distinct features of biomarker trajectories and define different biomarker-risk relationships. Results: Instantaneous forms capture the biomarker's current level or short-term dynamics, whereas cumulative and change-based forms reflect longer-term exposure or trends. Variability-based structures quantify instability in the biomarker trajectory as an alternative prognostic signal. Association parameters depend on the functional form, biomarker scale, and time scale, and effect sizes are therefore not directly comparable. In the MIRAGE application, alternative functional forms produced different effect interpretations and, in some cases, different conclusions regarding the biomarker-risk relationship. Conclusions: The choice of functional form is a key modelling decision in joint models and determines the interpretation of the biomarker-risk association. Aligning the functional form with the scientific question is essential for valid interpretation and transparent reporting.